Optimal. Leaf size=163 \[ \frac{1}{4} a^2 c x^4+\frac{1}{5} a^2 d x^5+\frac{1}{6} a^2 e x^6+\frac{1}{10} b x^{10} (2 a f+b c)+\frac{1}{7} a x^7 (a f+2 b c)+\frac{1}{11} b x^{11} (2 a g+b d)+\frac{1}{8} a x^8 (a g+2 b d)+\frac{1}{12} b x^{12} (2 a h+b e)+\frac{1}{9} a x^9 (a h+2 b e)+\frac{1}{13} b^2 f x^{13}+\frac{1}{14} b^2 g x^{14}+\frac{1}{15} b^2 h x^{15} \]
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Rubi [A] time = 0.15901, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.026, Rules used = {1820} \[ \frac{1}{4} a^2 c x^4+\frac{1}{5} a^2 d x^5+\frac{1}{6} a^2 e x^6+\frac{1}{10} b x^{10} (2 a f+b c)+\frac{1}{7} a x^7 (a f+2 b c)+\frac{1}{11} b x^{11} (2 a g+b d)+\frac{1}{8} a x^8 (a g+2 b d)+\frac{1}{12} b x^{12} (2 a h+b e)+\frac{1}{9} a x^9 (a h+2 b e)+\frac{1}{13} b^2 f x^{13}+\frac{1}{14} b^2 g x^{14}+\frac{1}{15} b^2 h x^{15} \]
Antiderivative was successfully verified.
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Rule 1820
Rubi steps
\begin{align*} \int x^3 \left (a+b x^3\right )^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx &=\int \left (a^2 c x^3+a^2 d x^4+a^2 e x^5+a (2 b c+a f) x^6+a (2 b d+a g) x^7+a (2 b e+a h) x^8+b (b c+2 a f) x^9+b (b d+2 a g) x^{10}+b (b e+2 a h) x^{11}+b^2 f x^{12}+b^2 g x^{13}+b^2 h x^{14}\right ) \, dx\\ &=\frac{1}{4} a^2 c x^4+\frac{1}{5} a^2 d x^5+\frac{1}{6} a^2 e x^6+\frac{1}{7} a (2 b c+a f) x^7+\frac{1}{8} a (2 b d+a g) x^8+\frac{1}{9} a (2 b e+a h) x^9+\frac{1}{10} b (b c+2 a f) x^{10}+\frac{1}{11} b (b d+2 a g) x^{11}+\frac{1}{12} b (b e+2 a h) x^{12}+\frac{1}{13} b^2 f x^{13}+\frac{1}{14} b^2 g x^{14}+\frac{1}{15} b^2 h x^{15}\\ \end{align*}
Mathematica [A] time = 0.027851, size = 163, normalized size = 1. \[ \frac{1}{4} a^2 c x^4+\frac{1}{5} a^2 d x^5+\frac{1}{6} a^2 e x^6+\frac{1}{10} b x^{10} (2 a f+b c)+\frac{1}{7} a x^7 (a f+2 b c)+\frac{1}{11} b x^{11} (2 a g+b d)+\frac{1}{8} a x^8 (a g+2 b d)+\frac{1}{12} b x^{12} (2 a h+b e)+\frac{1}{9} a x^9 (a h+2 b e)+\frac{1}{13} b^2 f x^{13}+\frac{1}{14} b^2 g x^{14}+\frac{1}{15} b^2 h x^{15} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 152, normalized size = 0.9 \begin{align*}{\frac{{b}^{2}h{x}^{15}}{15}}+{\frac{{b}^{2}g{x}^{14}}{14}}+{\frac{{b}^{2}f{x}^{13}}{13}}+{\frac{ \left ( 2\,abh+{b}^{2}e \right ){x}^{12}}{12}}+{\frac{ \left ( 2\,abg+{b}^{2}d \right ){x}^{11}}{11}}+{\frac{ \left ( 2\,abf+{b}^{2}c \right ){x}^{10}}{10}}+{\frac{ \left ({a}^{2}h+2\,aeb \right ){x}^{9}}{9}}+{\frac{ \left ({a}^{2}g+2\,bda \right ){x}^{8}}{8}}+{\frac{ \left ({a}^{2}f+2\,abc \right ){x}^{7}}{7}}+{\frac{{a}^{2}e{x}^{6}}{6}}+{\frac{{a}^{2}d{x}^{5}}{5}}+{\frac{{a}^{2}c{x}^{4}}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.935679, size = 204, normalized size = 1.25 \begin{align*} \frac{1}{15} \, b^{2} h x^{15} + \frac{1}{14} \, b^{2} g x^{14} + \frac{1}{13} \, b^{2} f x^{13} + \frac{1}{12} \,{\left (b^{2} e + 2 \, a b h\right )} x^{12} + \frac{1}{11} \,{\left (b^{2} d + 2 \, a b g\right )} x^{11} + \frac{1}{10} \,{\left (b^{2} c + 2 \, a b f\right )} x^{10} + \frac{1}{9} \,{\left (2 \, a b e + a^{2} h\right )} x^{9} + \frac{1}{6} \, a^{2} e x^{6} + \frac{1}{8} \,{\left (2 \, a b d + a^{2} g\right )} x^{8} + \frac{1}{5} \, a^{2} d x^{5} + \frac{1}{7} \,{\left (2 \, a b c + a^{2} f\right )} x^{7} + \frac{1}{4} \, a^{2} c x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.0839, size = 409, normalized size = 2.51 \begin{align*} \frac{1}{15} x^{15} h b^{2} + \frac{1}{14} x^{14} g b^{2} + \frac{1}{13} x^{13} f b^{2} + \frac{1}{12} x^{12} e b^{2} + \frac{1}{6} x^{12} h b a + \frac{1}{11} x^{11} d b^{2} + \frac{2}{11} x^{11} g b a + \frac{1}{10} x^{10} c b^{2} + \frac{1}{5} x^{10} f b a + \frac{2}{9} x^{9} e b a + \frac{1}{9} x^{9} h a^{2} + \frac{1}{4} x^{8} d b a + \frac{1}{8} x^{8} g a^{2} + \frac{2}{7} x^{7} c b a + \frac{1}{7} x^{7} f a^{2} + \frac{1}{6} x^{6} e a^{2} + \frac{1}{5} x^{5} d a^{2} + \frac{1}{4} x^{4} c a^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.13967, size = 167, normalized size = 1.02 \begin{align*} \frac{a^{2} c x^{4}}{4} + \frac{a^{2} d x^{5}}{5} + \frac{a^{2} e x^{6}}{6} + \frac{b^{2} f x^{13}}{13} + \frac{b^{2} g x^{14}}{14} + \frac{b^{2} h x^{15}}{15} + x^{12} \left (\frac{a b h}{6} + \frac{b^{2} e}{12}\right ) + x^{11} \left (\frac{2 a b g}{11} + \frac{b^{2} d}{11}\right ) + x^{10} \left (\frac{a b f}{5} + \frac{b^{2} c}{10}\right ) + x^{9} \left (\frac{a^{2} h}{9} + \frac{2 a b e}{9}\right ) + x^{8} \left (\frac{a^{2} g}{8} + \frac{a b d}{4}\right ) + x^{7} \left (\frac{a^{2} f}{7} + \frac{2 a b c}{7}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06638, size = 216, normalized size = 1.33 \begin{align*} \frac{1}{15} \, b^{2} h x^{15} + \frac{1}{14} \, b^{2} g x^{14} + \frac{1}{13} \, b^{2} f x^{13} + \frac{1}{6} \, a b h x^{12} + \frac{1}{12} \, b^{2} x^{12} e + \frac{1}{11} \, b^{2} d x^{11} + \frac{2}{11} \, a b g x^{11} + \frac{1}{10} \, b^{2} c x^{10} + \frac{1}{5} \, a b f x^{10} + \frac{1}{9} \, a^{2} h x^{9} + \frac{2}{9} \, a b x^{9} e + \frac{1}{4} \, a b d x^{8} + \frac{1}{8} \, a^{2} g x^{8} + \frac{2}{7} \, a b c x^{7} + \frac{1}{7} \, a^{2} f x^{7} + \frac{1}{6} \, a^{2} x^{6} e + \frac{1}{5} \, a^{2} d x^{5} + \frac{1}{4} \, a^{2} c x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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